Petrological Geodynamics of Mantle Melting I. AlphaMELTS + Multiphase Flow: Dynamic Equilibrium Melting, Method and Results

The complex process of melting in the Earth's interior is studied by combining a multiphase numerical flow model with the program AlphaMELTS which provides a petrological description based on thermodynamic principles. The objective is to address the fundamental question of the effect of the mantle and melt dynamics on the composition and abundance of the melt and the residual solid. The conceptual idea is based on a 1-D description of the melting process that develops along an ideal vertical column where local chemical equilibrium is assumed to apply at some level in space and time. By coupling together the transport model and the chemical thermodynamic model, the evolution of the melting process can be described in terms of melt distribution, temperature, pressure and solid and melt velocities but also variation of melt and residual solid composition and mineralogical abundance at any depth over time. In this first installment of a series of three contributions, a two-phase flow model (melt and solid assemblage) is developed under the assumption of complete local equilibrium between melt and a peridotitic mantle (dynamic equilibrium melting, DEM). The solid mantle is also assumed to be completely dry. The present study addresses some but not all the potential factors affecting the melting process. The influence of permeability and viscosity of the solid matrix are considered in some detail. The essential features of the dynamic model and how it is interfaced with AlphaMELTS are clearly outlined. A detailed and explicit description of the numerical procedure should make this type of numerical models less obscure. The general observation that can be made from the outcome of several simulations carried out for this work is that the melt composition varies with depth, however the melt abundance not necessarily always increases moving upwards. When a quasi-steady state condition is achieved, that is when melt abundance does not varies significantly with time, the melt and solid composition approach the composition that is found from a dynamic batch melting model which assumes the velocities of melt and residual solid to be the same. Time dependent melt fluctuations can be observed under certain conditions. In this case the composition of the melt that reaches the top side of the model (exit point) may vary to some extent. A consistent result of the model under various conditions is that the volume of the first melt that arrives at the exit point is substantially larger than any later melt output. The analogy with large magma emplacements associated to continental break-up or formation of oceanic plateaus seems to suggest that these events are the direct consequence of a dynamic two-phase flow process. Even though chemical equilibrium between melt and the residual solid is imposed locally in space, bulk composition of the whole system (solid+melt) varies with depth and may also vary with time, mainly as the result of the changes of the melt abundance. Potential factors that can influence the melting process such as bulk composition, temperature and mantle upwelling velocity at the top boundary (passive flow) or bottom boundary (active flow) should be addressed more systematically before the DEM model in this study and the dynamic fractional melting (DFM) model that will be introduced in the second installment can be applied to interpret real petrological data. Complete data files of most of the simulations and four animations are available following the data repository link provided in the Supplementary Material

Formulações numéricas conservativas para aproximação de modelos hiperbólicos com termos de fonte e problemas de transporte relacionados

Orientador: Eduardo Cardoso de AbreuTese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação CientíficaResumo: O objetivo desta tese é desenvolver, pelo menos no aspecto formal, algoritmos construtivos e bem-balanceados para a aproximação de classes específicas de modelos diferenciais. Nossas principais aplicações consistem em equações de água rasa e problemas de convecção-difusão no contexto de fenômenos de transporte, relacionados a problemas de pressão capilar descontínua em meios porosos. O foco principal é desenvolver sob o framework Lagrangian-Euleriano um esquema simples e eficiente para, em nível discreto, levar em conta o delicado equilíbrio entre as aproximações numéricas não lineares do fluxo hiperbólico e o termo fonte, e entre o fluxo hiperbólico e o operador difusivo. Os esquemas numéricos são propostos para ser independentes de estruturas particulares das funções de fluxo. Apresentamos diferentes abordagens que selecionam a solução entrópica qualitativamente correta, amparados por um grande conjunto de experimentos numéricos representativosAbstract: The purpose of this thesis is to develop, at least formally by construction, conservative methods for approximating specific classes of differential models. Our major applications consist in shallow water equations and nonstandard convection-diffusion problems in the context of transport phenomena, related to discontinuous capillary pressure problems in porous media. The main focus in this work is to develop under the Lagrangian-Eulerian framework a simple and efficient scheme to, on the discrete level, account for the delicate nonlinear balance between the numerical approximations of the hyperbolic flux and source term, and between the hyperbolic flux and the diffusion operator. The proposed numerical schemes are aimed to be independent of particular structures of the flux functions. We present different approaches that select the qualitatively correct entropy solution, supported by a large set of representative numerical experimentsDoutoradoMatematica AplicadaDoutor em Matemática Aplicada165564/2014-8CNPQCAPE

Numerical Modeling Of Degenerate Equations In Porous Media Flow: Degenerate Multiphase Flow Equations In Porous Media

In this paper is introduced a new numerical formulation for solving degenerate nonlinear coupled convection dominated parabolic systems in problems of flow and transport in porous media by means of a mixed finite element and an operator splitting technique, which, in turn, is capable of simulating the flow of a distinct number of fluid phases in different porous media regions. This situation naturally occurs in practical applications, such as those in petroleum reservoir engineering and groundwater transport. To illustrate the modelling problem at hand, we consider a nonlinear three-phase porous media flow model in one- and two-space dimensions, which may lead to the existence of a simultaneous one-, two- and three-phase flow regions and therefore to a degenerate convection dominated parabolic system. Our numerical formulation can also be extended for the case of three space dimensions. As a consequence of the standard mixed finite element approach for this flow problem the resulting linear algebraic system is singular. By using an operator splitting combined with mixed finite element, and a decomposition of the domain into different flow regions, compatibility conditions are obtained to bypass the degeneracy in order to the degenerate convection dominated parabolic system of equations be numerically tractable without any mathematical trick to remove the singularity, i.e., no use of a parabolic regularization. Thus, by using this procedure, we were able to write the full nonlinear system in an appropriate way in order to obtain a nonsingular system for its numerical solution. The robustness of the proposed method is verified through a large set of high-resolution numerical experiments of nonlinear transport flow problems with degenerating diffusion conditions and by means of a numerical convergence study. © Springer Science+Business Media New York 2012.553688717Aarnes, J.E., Krogstad, S., Lie, K.-A., A hierarchical multiscale method for two-phase flow based upon mixed finite elements and nonuniform coarse grids (2006) Multiscale Model. Simul., 5 (2), pp. 337-363Aarnes, J.E., Hauge, V.L., Efendiev, Y., Coarsening of three-dimensional structured and unstructured grids for subsurface flow (2007) Adv. Water Resour., 30 (11), pp. 2177-2193Aarnes, J.E., Krogstad, S., Lie, K.-A., Multiscale mixed/mimetic methods on corner-point grids (2008) Comput. Geosci., 12, pp. 297-315Abreu, E., Furtado, F., Pereira, F., On the numerical simulation of three-phase reservoir transport problem (2004) Transp. Theory Stat. Phys., 33, pp. 1-24Abreu, E., Douglas, J., Furtado, F., Marchesin, D., Pereira, F., Three-phase immiscible displacement in heterogeneous petroleum reservoirs (2006) Math. Comput. Simul., 73, pp. 2-20Abreu, E., Douglas, J., Furtado, F., Pereira, F., Operator splitting based on physics for flow in porous media (2008) Int. J. Comput. Sci., 2, pp. 315-335Azevedo, A., Marchesin, D., Plohr, B.J., Zumbrun, K., Capillary instability in models for three-phase flow (2002) Z. Angew. Math. Phys., 53, pp. 713-746Azevedo, A., Souza, A., Furtado, F., Marchesin, D., Plohr, B., The solution by the wave curve method of three-phase flow in virgin reservoirs (2010) Transp. Porous Media, 83, pp. 99-125Blunt, M.J., Thiele, M.R., Batycky, R.P., A 3D field scale streamline-based reservoir simulator (1997) SPE Reserv. Eng., pp. 246-254Borges, M.R., Furtado, F., Pereira, F., Amaral Souto, H.P., Scaling analysis for the tracer flow problem in self-similar permeability fields multiscale model (2008) Multiscale Model. Simul., 7, pp. 1130-1147Brooks, R.H., Corey, A.T., Hydraulic properties of porous media (1964) Hydrology Paper No. 3, pp. 1-27. , Colorado State University, Fort CollinsBürger, R., Coronel, A., Sepúlveda, M., A semi-implicit monotone difference scheme for an initialboundary value problem of a strongly degenerate parabolic equation modelling sedimentation-consolidation processes (2006) Math. Comput., 75, pp. 91-112Cavalli, F., Naldi, G., Puppo, G., Semplice, M., High-order relaxation schemes for nonlinear degenerate diffusion problems (2007) SIAM J. Numer. Anal., 45, pp. 2098-2119Chavent, G., Jaffré, J., Mathematical models and finite elements for reservoir simulation (1986) Studies in Applied Mathematics, 17. , North-Holland, AmsterdamChen, Z., Ewing, R.E., Comparison of various formulation of three-phase flow in porous media (1997) J. Comput. Phys., 132, pp. 362-373Chertock, A., Doering, C.R., Kashdan, E., Kurganov, A., A fast explicit operator splitting method for passive scalar advection (2010) J. Sci. Comput., 45, pp. 200-214Chrispella, J.C., Ervin, V.J., Jenkinsa, E.W., A fractional step θ-method for convection-diffusion problems (2007) J. Math. Anal. Appl., 333, pp. 204-218Corey, A., Rathjens, C., Henderson, J., Wyllie, M., Three-phase relative permeability (1956) Trans. Am. Inst. Min. Metall. Eng., 207, pp. 349-351Dafermos, C.M., (2010) Hyperbolic Conservation Laws in Continuum Physics, , 3rd edn. Springer, BerlinDahle, H.K., Ewing, R.E., Russell, T.F., Eulerian-lagrangian localized adjoint methods for a nonlinear advection-diffusion equation (1995) Comput. Methods Appl. Mech. Eng., 122, pp. 223-250Di Chiara Roupert, R., Chavent, G., Schafer, G., Three-phase compressible flow in porous media: Total differential compatible interpolation of relative permeabilities (2010) J. Comput. Phys., 229, pp. 4762-4780Doughty, C., Pruess, K., Modeling supercritical carbon dioxide injection in heterogeneous porous media (2004) Vadose Zone J., 3, pp. 837-847Douglas Jr., J., Furtado, F., Pereira, F., On the numerical simulation of waterflooding of heterogeneous petroleum reservoirs (1997) Comput. Geosci., 1, pp. 155-190Douglas, J., Russell, T.F., Numerical methods for convection-dominated diffusion problems based on combining the method of characteristics with finite element or finite difference procedures (1982) SIAM J. Numer. Anal., 19 (5), pp. 871-885Douglas, J., Pereira, F., Yeh, L.-M., A locally conservative eulerian-lagrangian method for flow in a porous medium of a mixture of two components having different densities (2000) Numerical Treatment of Multiphase Flows in Porous Media. Lecture Notes in Physics, 552, pp. 138-155. , Springer, BerlinDria, D.E., Pope, G.A., Sepehrnoori, K., Three-phase gas/oil/brine relative permeabilities measured under CO 2 flooding conditions (1993) Soc. Pet. Eng. J., pp. 143-150. , 20184Durlofsky, L.J., A triangle based mixed finite element-finite volume technique for modeling two phase flow through porous media (1993) J. Comput. Phys., 105, pp. 252-266Durlofsky, L.J., Accuracy of mixed and control volume finite element approximations to darcy velocity and related quantities (1994) Water Resour. Res., 30, pp. 965-973Durlofsky, L.J., Upscaling and gridding of fine scale geological models for flow simulation (2005) 8th International Forum on Reservoir Simulation Iles Borromees, , Presented at Italy, June 20-24Durlofsky, L.J., Jones, R.C., Milliken, W.J., A nonuniform coarsening approach for the scale-up of displacement processes in heterogeneous porous media (1997) Adv. Water Resour., 20, pp. 335-347Fortin, M., Brezzi, F., (1991) Mixed and Hybrid Finite Element Methods, , In: Springer Series in Computational Mathematics. Springer, BerlinFurtado, F., Pereira, F., Crossover from nonlinearity controlled to heterogeneity controlled mixing in two-phase porous media flows (2003) Comput. Geosci., 7, pp. 115-135Gasda, S.E., Farthing, M.W., Kees, C.E., Miller, C.T., Adaptive split-operator methods for modeling transport phenomena in porous medium systems (2011) Adv. Water Resour., 34, pp. 1268-1282Gerritsen, M.G., Durlofsky, L.J., Modeling fluid flow in oil reservoirs (2006) Annu. Rev. 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Numerical Modelling Of Three-phase Immiscible Flow In Heterogeneous Porous Media With Gravitational Effects

This paper presents a new numerical formulation for the simulation of immiscible and incompressible three-phase water-gas-oil flows in heterogeneous porous media. We take into account the gravitational effects, both variable permeability and porosity of porous medium, and explicit spatially varying capillary pressure, in the diffusive fluxes, and explicit spatially varying flux functions, in the hyperbolic operator. The new formulation is a sequential time marching fractional-step procedure based in a splitting technique to decouple the equations with mixed discretization techniques for each of the subproblems: convection, diffusion, and pressure-velocity. The system of nonlinear hyperbolic equations that models the convective transport of the fluid phases is approximated by a modified central scheme to take into account the explicit spatially discontinuous flux functions and the effects of spatially variable porosity. This scheme is coupled with a locally conservative mixed finite element formulation for solving parabolic and elliptic problems, associated respectively with the diffusive transport of fluid phases and the pressure-velocity problem. The time discretization of the parabolic problem is performed by means of an implicit backward Euler procedure. The hybrid-mixed formulation reported here is designed to handle discontinuous capillary pressures. The new method is used to numerically investigate the question of existence, and structurally stable, of three-phase flow solutions for immiscible displacements in heterogeneous porous media with gravitational effects. Our findings appear to be consistent with theoretical and experimental results available in the literature. © 2013 IMACS.97234259Abreu, E., Conceição, D., Numerical Coupling of Multiphysics Flows in Porous Media, 3rd International Conference on Approximation Methods and numerical Modeling in Environment and Natural Resources, Pau, France (2009) Proceedings of MAMERN, 1, pp. 19-24Abreu, E., Douglas, Jr.J., Furtado, F., Pereira, F., Operator splitting based on physics for flow in porous media (2008) International Journal of Computational Science, 2, pp. 315-335Abreu, E., Douglas, Jr.J., Furtado, F., Pereira, F., Operator splitting for three-phase flow in heterogeneous porous media (2008) Communications in Computational Physics, 6, pp. 72-84Abreu, E., Douglas Jr., J., Furtado, F., Marchesin, D., Pereira, F., Three-phase immiscible displacement in heterogeneous petroleum reservoirs (2006) Mathematics and Computers in Simulation, 73 (1-4 SPEC. 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ShearerVan Leer, B., Towards the ultimate conservative difference scheme v - A second order sequel to Godunov's method (1979) Journal of Computational Physics, 32, pp. 101-136Zhou, D., Blunt, M., Wettability effects in three-phase gravity drainage (1998) Journal of Petroleum Science and Engineering, 20 (3-4), pp. 203-211. , DOI 10.1016/S0920-4105(98)00021-7, PII S092041059800021